Eco math book (1)

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Chapter 2

Equations. Functions of one variable.

Complex numbers

2.1 ax2 + bx + c = 0 ⇐⇒ x1,2 =−b±√b2 − 4ac

2a

The roots of the gen-eral quadratic equation.They are real providedb2 ≥ 4ac (assuming thata, b, and c are real).

2.2

If x1 and x2 are the roots of x2 + px + q = 0,then

x1 + x2 = −p, x1x2 = q

Viete’s rule.

2.3 ax3 + bx2 + cx + d = 0 The general cubicequation.

2.4 x3 + px + q = 0(2.3) reduces to the form(2.4) if x in (2.3) isreplaced by x − b/3a.

2.5

x3 + px + q = 0 with Δ = 4p3 + 27q2 has

• three different real roots if Δ < 0;

• three real roots, at least two of which areequal, if Δ = 0;

• one real and two complex roots if Δ > 0.

Classification of theroots of (2.4) (assumingthat p and q are real).

2.6

The solutions of x3 + px + q = 0 are

x1 = u + v, x2 = ωu + ω2v, and x3 = ω2u + ωv,

where ω = −12

+ i2

√3, and

u =3 −q2 + 12

4p

3

+ 27q2

27

v =3

−q

2− 1

2

4p3 + 27q2

27

Cardano’s formulas

for the roots of a cubicequation. i is the imagi-nary unit (see (2.75))and ω is a complex thirdroot of 1 (see (2.88)).

(If complex numbers be-come involved, the cuberoots must be chosen sothat 3uv = −p. Don’ttry to use these formulasunless you have to!)



8

2.7

If x1, x2, and x3 are the roots of the equationx3 + px2 + qx + r = 0, then

x1 + x2 + x3 =

−p

x1x2 + x1x3 + x2x3 = q

x1x2x3 = −r

Useful relations.

2.8 P (x) = anxn + an−1xn−1 + · · ·+ a1x + a0A polynomial of degreen. (an = 0.)

2.9

For the polynomial P (x) in (2.8) there existconstants x1, x2, . . . , xn (real or complex) such

thatP (x) = an(x− x1) · · · (x− xn)

The fundamental

theorem of algebra .x1, . . . , xn are called

zeros of P (x) and rootsof P (x) = 0.

2.10

x1 + x2 + · · ·+ xn = −an−1an

x1x2 + x1x3 + · · ·+ xn−1xn =i<j

xixj =an−2

an

x1x2 · · ·xn = (−1)na0an

Relations between theroots and the coefficientsof P (x) = 0, where P (x)is defined in (2.8). (Gen-eralizes (2.2) and (2.7).)

2.11

If an−1, . . . , a1, a0 are all integers, then anyinteger root of the equation

xn + an−1xn−1 + · · ·+ a1x + a0 = 0

must divide a0.

Any integer solutions of x3 + 6x2 − x − 6 = 0must divide −6. (In thiscase the roots are ±1and −6.)

2.12

Let k be the number of changes of sign in thesequence of coefficients an, an−1, . . . , a1, a0

in (2.8). The number of positive real roots of P (x) = 0, counting the multiplicities of theroots, is k or k minus a positive even number.If k = 1, the equation has exactly one positivereal root.

Descartes’s rule of signs.

2.13

The graph of the equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

is

• an ellipse, a point or empty if 4AC > B2;

• a parabola, a line, two parallel lines, orempty if 4AC = B2;

• a hyperbola or two intersecting lines if 4AC < B2.

Classification of conics.A, B, C not all 0.



9

2.14x = x cos θ − y sin θ, y = x sin θ + y cos θ

with cot 2θ = (A

−C )/B

Transforms the equa-tion in (2.13) into aquadratic equation in

x

and y

, where thecoefficient of xy is 0.

2.15 d =

(x2 − x1)2 + (y2 − y1)2The (Euclidean) distance

between the points(x1, y1) and (x2, y2).

2.16 (x− x0)2 + (y − y0)2 = r2Circle with center at(x0, y0) and radius r.

2.17(x− x0)2

a2+

(y − y0)2

b2= 1

Ellipse with center at(x0, y0) and axes parallelto the coordinate axes.

2.18

y

x

r(x, y)

x0

y0

y

x

b

a

(x, y)

x0

y0 Graphs of (2.16) and(2.17).

2.19(x− x0)2

a2− (y − y0)2

b2= ±1

Hyperbola with center at(x0, y0) and axes parallelto the coordinate axes.

2.20Asymptotes for (2.19):

y − y0 = ± b

a(x− x0)

Formulas for asymp-totes of the hyperbolasin (2.19).

2.21

y

xx0

y0 ab

y

x

ba

x0

y0

Hyperbolas with asymp-totes, illustrating (2.19)and (2.20), correspond-ing to + and − in(2.19), respectively. Thetwo hyperbolas have thesame asymptotes.

2.22 y − y0 = a(x− x0)2

, a = 0

Parabola with vertex

(x0, y0) and axis parallelto the y-axis.

2.23 x− x0 = a(y − y0)2, a = 0Parabola with vertex(x0, y0) and axis parallelto the x-axis.



10

2.24

y

x

y0

x0

y

x

y0

x0

Parabolas illustrating

(2.22) and (2.23) witha > 0.

2.25

A function f is

• increasing if

x1 < x2 ⇒ f (x1) ≤ f (x2)

• strictly increasing if x1 < x2 ⇒ f (x1) < f (x2)

• decreasing if

x1 < x2 ⇒ f (x1) ≥ f (x2)

• strictly decreasing if

x1 < x2 ⇒ f (x1) > f (x2)

• even if f (x) = f (−x) for all x

•odd if f (x) =

−f (

−x) for all x

• symmetric about the line x = a if

f (a + x) = f (a− x) for all x

• symmetric about the point (a, 0) if

f (a− x) = −f (a + x) for all x

• periodic (with period k) if there exists anumber k > 0 such that

f (x + k) = f (x) for all x

Properties of functions.

2.26

• If y = f (x) is replaced by y = f (x) + c, thegraph is moved upwards by c units if c > 0(downwards if c is negative).

• If y = f (x) is replaced by y = f (x + c), thegraph is moved c units to the left if c > 0 (tothe right if c is negative).

•If y = f (x) is replaced by y = cf (x), the

graph is stretched vertically if c > 0 (stretch-ed vertically and reflected about the x-axisif c is negative).

• If y = f (x) is replaced by y = f (−x), thegraph is reflected about the y-axis.

Shifting the graph of y = f (x).



11

2.27

y

x

y

x

Graphs of increasingand strictly increasingfunctions.

2.28

y

x

y

x

Graphs of decreasingand strictly decreasingfunctions.

2.29

y

x

y

x

y

xx = a

Graphs of even and oddfunctions, and of a func-tion symmetric aboutx = a.

2.30

y

x(a, 0)

y

x

kGraphs of a functionsymmetric about thepoint (a, 0) and of afunction periodic withperiod k.

2.31

y = ax + b is a nonvertical asymptote for thecurve y = f (x) if

limx→∞f (x)

−(ax + b) = 0

or

limx→−∞

f (x)− (ax + b)

= 0

Definition of a nonverti-

cal asymptote.

2.32

y

x

f (x) − (ax + b)

y = ax + b

y = f (x)

x

y = ax + b is anasymptote for the curvey = f (x).



12

2.33

How to find a nonvertical asymptote for thecurve y = f (x) as x →∞:

•Examine lim

x→∞f (x)/x. If the limit does not

exist, there is no asymptote as x →∞.

• If limx→∞

f (x)/x

= a, examine the limit

limx→∞

f (x)−ax

. If this limit does not exist,

the curve has no asymptote as x →∞.

• If limx→∞

f (x)−ax

= b, then y = ax + b is an

asymptote for the curve y = f (x) as x →∞.

Method for finding non-vertical asymptotes fora curve y = f (x) asx → ∞. Replacingx → ∞ by x → −∞gives a method for find-ing nonvertical asymp-totes as x → −∞.

2.34

To find an approximate root of f (x) = 0, definexn for n = 1, 2, . . . , by

xn+1 = xn − f (xn)

f (xn)

If x0 is close to an actual root x∗, the sequence{xn} will usually converge rapidly to that root.

Newton’s approxima-

tion method . (A rule of thumb says that, to ob-tain an approximationthat is correct to n deci-mal places, use Newton’smethod until it gives thesame n decimal placestwice in a row.)

2.35

y

xxn xn+1

x∗

y = f (x)

Illustration of Newton’sapproximation method.The tangent to thegraph of f at (xn, f (xn))intersects the x-axis atx = xn+1.

2.36

Suppose in (2.34) that f (x∗) = 0, f (x∗)

= 0,

and that f (x∗) exists and is continuous in aneighbourhood of x∗. Then there exists a δ > 0such that the sequence {xn} in (2.34) convergesto x∗ when x0 ∈ (x∗ − δ, x∗ + δ).

Sufficient conditions forconvergence of Newton’smethod.

2.37

Suppose in (2.34) that f is twice differentiablewith f (x∗) = 0 and f (x∗) = 0. Suppose fur-ther that there exist a K > 0 and a δ > 0 suchthat for all x in (x∗ − δ, x∗ + δ),

|f (x)f

(x)|f (x)2

≤ K |x− x∗| < 1

Then if x0 ∈ (x∗−δ, x∗+ δ), the sequence {xn}in (2.34) converges to x∗ and

|xn − x∗| ≤ (δK )2n

/K

A precise estimation of

the accuracy of Newton’smethod.



13

2.38 y − f (x1) = f (x1)(x− x1)The equation for thetangent to y = f (x) at(x1, f (x1)).

2.39 y − f (x1) = − 1

f (x1)(x− x1)

The equation for thenormal to y = f (x) at(x1, f (x1)).

2.40

y

xx1

y = f (x)

tangentnormal

The tangent and thenormal to y = f (x) at(x1, f (x1)).

2.41

(i) ar · as = ar+s (ii) (ar)s = ars

(iii) (ab)r = arbr (iv) ar/as = ar−s

(v)a

b

r=

ar

br(vi) a−r =

1

ar

Rules for powers. (r ands are arbitrary real num-bers, a and b are positivereal numbers.)

2.42

• e = limn→∞

1 + 1

nn

= 2.718281828459 . . .

• ex = limn→∞

1 +

x

n

n• lim

n→∞an = a ⇒ lim

n→∞

1 +

ann

n= ea

Important definitionsand results. See (8.23)for another formula forex.

2.43 elnx = xDefinition of the naturallogarithm.

2.44

y

x

ln x

ex

1

1

The graphs of y = ex

and y = ln x are sym-metric about the liney = x.

2.45

ln(xy) = ln x + ln y; lnx

y

= ln x

−ln y

ln xp = p ln x; ln1

x= − ln x

Rules for the natural

logarithm function.(x and y are positive.)

2.46 aloga x = x (a > 0, a = 1)Definition of the loga-

rithm to the base a.



14

2.47loga x =

ln x

ln a; loga b · logb a = 1

loge x = ln x; log10 x = log10 e

·ln x

Logarithms with differ-ent bases.

2.48

loga(xy) = loga x + loga y

logax

y= loga x− loga y

loga xp = p loga x, loga1

x= − loga x

Rules for logarithms.(x and y are positive.)

2.49 1◦ =π

180

rad, 1 rad = 180

π◦

Relationship between de-grees and radians (rad).

2.50 0

π/6π/4

π/3π/2

3π/4

π

3π/2

0◦

90◦

180◦

270◦

30◦45◦

60◦

135◦

Relations between de-grees and radians.

2.51

x1

cosx

sin xtanx

cotx

Definitions of the basictrigonometric functions.x is the length of the

arc, and also the radianmeasure of the angle.

2.52

y

x

−3π

2 −π

−π

2

π

2π 3π

2

y = sinxy = cos x

The graphs of y = sin x(—) and y = cos x (---).The functions sin andcos are periodic withperiod 2π:

sin(x + 2π) = sin x,cos(x + 2π) = cos x.

2.53 tan x =sin x

cos x, cot x =

cos x

sin x=

1

tan xDefinition of the tangent

and cotangent functions.



15

2.54

y

x

− 3π

2−π −π

2

π

2π 3π

2

y = tan xy = cot x

The graphs of y = tan x

(—) and y = cot x (---).The functions tan andcot are periodic withperiod π:tan(x + π) = tan x,cot(x + π) = cot x.

2.55

x 0 π

6

= 30◦ π

4

= 45◦ π

3

= 60◦ π

2

= 90◦

sin x 0 1

2

1

2

√ 2 1

2

√ 3 1

cosx 1 1

2

√ 3 1

2

√ 2 1

20

tanx 0 1

3

√ 3 1

√ 3 ∗

cotx ∗ √ 3 1 1

3

√ 3 0

* not defined

Special values of thetrigonometric functions.

2.56

x 3π

4 = 135◦

π = 180◦

3π

2 = 270◦

2π = 360◦

sin x 1

2

√ 2 0 −1 0

cosx −1

2

√ 2 −1 0 1

tanx −1 0 ∗ 0

cot x −1 ∗ 0 ∗

* not defined

2.57 limx→0

sin axx

= a An important limit.

2.58 sin2 x + cos2 x = 1

Trigonometric formulas.(For series expansions of trigonometric functions,see Chapter 8.)

2.59 tan2 x =1

cos2 x− 1, cot2 x =

1

sin2 x− 1

2.60

cos(x + y) = cos x cos y − sin x sin y

cos(x− y) = cos x cos y + sin x sin y

sin(x + y) = sin x cos y + cos x sin y

sin(x− y) = sin x cos y − cos x sin y



16

2.61

tan(x + y) =tan x + tan y

1− tan x tan y

tan(x−

y) =tan x− tan y

1 + tan x tan y

Trigonometric formulas.

2.62cos2x = 2 cos2 x− 1 = 1− 2sin2 x

sin2x = 2 sin x cos x

2.63 sin2x

2=

1− cos x

2, cos2

x

2=

1 + cos x

2

2.64

cos x + cos y = 2 cosx + y

2

cosx− y

2cos x− cos y = −2sin

x + y

2sin

x− y

2

2.65sin x + sin y = 2 sin

x + y

2cos

x− y

2

sin x− sin y = 2 cosx + y

2sin

x− y

2

2.66

y = arcsin x

⇔x = sin y, x

∈[

−1, 1], y

∈[

−π

2

,π

2

]

y = arccos x ⇔ x = cos y, x ∈ [−1, 1], y ∈ [0, π]

y = arctan x ⇔ x = tan y, x ∈ R, y ∈ (−π

2,

π

2)

y = arccot x ⇔ x = cot y, x ∈ R, y ∈ (0, π)

Definitions of the inversetrigonometric functions.

2.67

y

x

y = arcsin x

−π

2

π

2

1−1

y

x

y = arccos x

1−1

π

π

2Graphs of the inverse

trigonometric functionsy = arcsin x and y =arccos x.

2.68

y

x

y = arccot x

y = arctan x

π

π

2

−π

2

1

Graphs of the inverse

trigonometric functionsy = arctan x and y =arccot x.



17

2.69arcsin x = sin−1 x, arccos x = cos−1 x

arctan x = tan−1 x, arccot x = cot−1 x

Alternative notation forthe inverse trigonometricfunctions.

2.70

arcsin(−x) = − arcsin x

arccos(−x) = π − arccos x

arctan(−x) = arctan x

arccot(−x) = π − arccot x

arcsin x + arccos x =π

2

arctan x + arccot x =π

2arctan

1

x=

π

2− arctan x, x > 0

arctan1

x= −π

2− arctan x, x < 0

Properties of the inversetrigonometric functions.

2.71 sinh x =ex − e−x

2, cosh x =

ex + e−x

2Hyperbolic sine andcosine.

2.72

y

x

y = sinh x

y = cosh x

1

1

Graphs of the hyperbolicfunctions y = sinh x andy = cosh x.

2.73

cosh2 x− sinh2 x = 1

cosh(x + y) = cosh x cosh y + sinh x sinh y

cosh2x = cosh2 x + sinh2 x

sinh(x + y) = sinh x cosh y + cosh x sinh y

sinh2x = 2sinh x cosh x

Properties of hyperbolicfunctions.

2.74

y = arsinh x ⇐⇒ x = sinh yy = arcosh x, x ≥ 1 ⇐⇒ x = cosh y, y ≥ 0

arsinh x = ln

x +

x2 + 1

arcosh x = ln

x +

x2 − 1

, x ≥ 1

Definition of the inversehyperbolic functions.



18

Complex numbers

2.75 z = a + ib, z = a− ib

A complex number and

its conjugate . a, b ∈ R,and i2 = −1. i is calledthe imaginary unit .

2.76 |z| =√

a2 + b2, Re(z) = a, Im(z) = b

|z| is the modulus of z = a + ib. Re(z) andIm(z) are the real andimaginary parts of z.

2.77

|z|

z = a + ib

z = a− ib

Real axis

Imaginary axis

a

b

Geometric representationof a complex numberand its conjugate.

2.78

• (a + ib) + (c + id) = (a + c) + i(b + d)

• (a + ib)− (c + id) = (a− c) + i(b− d)

• (a + ib)(c + id) = (ac− bd) + i(ad + bc)

• a + ib

c + id=

1

c2 + d2

(ac + bd) + i(bc− ad)

Addition, subtraction,

multiplication , anddivision of complexnumbers.

2.79 |z1

|=

|z1

|, z1z1 =

|z1

|2, z1 + z2 = z1 + z2,

|z1z2| = |z1||z2|, |z1 + z2| ≤ |z1|+ |z2|Basic rules. z1 and z2

are complex numbers.

2.80z = a + ib = r(cos θ + i sin θ) = reiθ, where

r = |z| =√

a2 + b2, cos θ =a

r, sin θ =

b

r

The trigonometric orpolar form of a complexnumber. The angle θ iscalled the argument of z.See (2.84) for eiθ.

2.81θ

r

a + ib = r(cos θ + i sin θ)

b

Imaginary axis

a Real axis

Geometric representa-tion of the trigonomet-ric form of a complexnumber.



19

2.82

If zk = rk(cos θk + i sin θk), k = 1, 2, then

z1z2 = r1r2

cos(θ1 + θ2) + i sin(θ1 + θ2)

z1

z2=

r1

r2

cos(θ1 − θ2) + i sin(θ1 − θ2)

Multiplication and di-vision on trigonometric

form.

2.83 (cos θ + i sin θ)n = cos nθ + i sin nθDe Moivre’s formula ,n = 0, 1, . . . .

2.84

If z = x + iy, then

ez = ex+iy = ex · eiy = ex(cos y + i sin y)

In particular,

eiy = cos y + i sin y

The complex exponential

function .

2.85 eπi = −1 A striking relationship.

2.86ez = ez, ez+2πi = ez, ez1+z2 = ez1ez2 ,

ez1−z2 = ez1/ez2Rules for the complexexponential function.

2.87 cos z =eiz + e−iz

2, sin z =

eiz − e−iz

2iEuler’s formulas.

2.88

If a = r(cos θ + i sin θ) = 0, then the equation

zn = a

has exactly n roots, namely

zk = n

√r

cosθ + 2kπ

n+ i sin

θ + 2kπ

n

for k = 0, 1, . . . , n− 1.

nth roots of a complexnumber, n = 1, 2, . . . .

References

Most of these formulas can be found in any calculus text, e.g. Edwards and Penney(1998) or Sydsæter and Hammond (2005). For (2.3)–(2.12), see e.g. Turnbull (1952).



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Eco math book (1)